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Properties

Label 2-1470-21.20-c1-0-28
Degree $2$
Conductor $1470$
Sign $0.533 - 0.845i$
Analytic cond. $11.7380$
Root an. cond. $3.42607$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + (−0.683 + 1.59i)3-s − 4-s − 5-s + (−1.59 − 0.683i)6-s − i·8-s + (−2.06 − 2.17i)9-s − i·10-s − 2.78i·11-s + (0.683 − 1.59i)12-s + 3.85i·13-s + (0.683 − 1.59i)15-s + 16-s + 6.13·17-s + (2.17 − 2.06i)18-s − 7.77i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.394 + 0.918i)3-s − 0.5·4-s − 0.447·5-s + (−0.649 − 0.279i)6-s − 0.353i·8-s + (−0.688 − 0.725i)9-s − 0.316i·10-s − 0.839i·11-s + (0.197 − 0.459i)12-s + 1.06i·13-s + (0.176 − 0.410i)15-s + 0.250·16-s + 1.48·17-s + (0.512 − 0.486i)18-s − 1.78i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.533 - 0.845i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.533 - 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.145559242\)
\(L(\frac12)\) \(\approx\) \(1.145559242\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.683 - 1.59i)T \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + 2.78iT - 11T^{2} \)
13 \( 1 - 3.85iT - 13T^{2} \)
17 \( 1 - 6.13T + 17T^{2} \)
19 \( 1 + 7.77iT - 19T^{2} \)
23 \( 1 + 5.94iT - 23T^{2} \)
29 \( 1 - 0.293iT - 29T^{2} \)
31 \( 1 - 9.46iT - 31T^{2} \)
37 \( 1 + 5.38T + 37T^{2} \)
41 \( 1 - 2.06T + 41T^{2} \)
43 \( 1 - 5.99T + 43T^{2} \)
47 \( 1 - 4.88T + 47T^{2} \)
53 \( 1 + 0.393iT - 53T^{2} \)
59 \( 1 - 2.81T + 59T^{2} \)
61 \( 1 + 0.167iT - 61T^{2} \)
67 \( 1 - 9.00T + 67T^{2} \)
71 \( 1 + 8.46iT - 71T^{2} \)
73 \( 1 + 9.46iT - 73T^{2} \)
79 \( 1 + 0.358T + 79T^{2} \)
83 \( 1 - 7.38T + 83T^{2} \)
89 \( 1 + 0.493T + 89T^{2} \)
97 \( 1 - 18.4iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.357692758307856184950867146958, −8.918579255149630246356792666462, −8.159228320350285336790258726155, −7.01944228754355709323359194759, −6.42107014177960858126455053691, −5.37796047358702186608999427472, −4.75471847277956955304943676347, −3.84556669502807242325658617963, −2.93738926701956040998557573355, −0.67816219584290587372632438172, 0.925846228721370166294544132123, 1.98595340164034972120638848574, 3.18870473676669420729341196973, 4.08833980897436012409710353419, 5.50528014713511996229281396124, 5.75431856468945551567486379367, 7.24608595514407029913232155244, 7.77955699563060158376583236502, 8.319579297935764113053551298909, 9.683998303567054886753760952319

Graph of the $Z$-function along the critical line